Mathematical Model of Call Center in the Form of Multi-Server Queueing System

The paper considers the model of a call center in the form of a multi-server queueing system with Poisson arrivals and an unlimited waiting area. In the model under consideration, incoming calls do not differ in terms of service conditions, requested service, and interarrival periods. It is assumed that an incoming call can use any free server and they are all identical in terms of capabilities and quality. The goal problem is to find the stationary distribution of the number of calls in the system for an arbitrary recurrent service. This will allow us to evaluate the performance measures of such systems and solve various optimization problems for them. Considering models with non-exponential service times provides solutions for a wide class of mathematical models, making the results more adequate for real call centers. The solution is based on the approximation of the given distribution function of the service time by the hyperexponential distribution function. Therefore, first, the problem of studying a system with hyperexponential service is solved using the matrix-geometric method. Further, on the basis of this result, an approximation of the stationary distribution of the number of calls in a multi-server system with an arbitrary distribution function of the service time is constructed. Various issues in the application of this approximation are considered, and its accuracy is analyzed based on comparison with the known analytical result for a particular case, as well as with the results of the simulation.

On the stability of queues with the dropping function

In this paper, the stability of the queueing system with the dropping function is studied. In such system, every incoming job may be dropped randomly, with the probability being a function of the queue length. The main objective of the work is to find an easy to use condition, sufficient for the instability of the system, under assumption of Poisson arrivals and general service time distribution. Such condition is found and proven using a boundary for the dropping function and analysis of the embedded Markov chain. Applicability of the proven condition is demonstrated on several examples of dropping functions. Additionally, its correctness is confirmed using a discrete-event simulator.

Amplification and attenuation of noisy expression by export processes

Inside mammalian cells, single genes are known to be transcribed in stochastic bursts leading to the synthesis of nuclear RNAs that are subsequently exported to the cytoplasm to create mRNAs. We systematically characterize the role of export processes in shaping the extent of random fluctuations (i.e. noise) in the mRNA level of a given gene. Using the method of Partitioning of Poisson arrivals, we derive an exact analytical expression for the noise in mRNA level assuming that the nuclear retention time of each RNA is an independent and identically distributed random variable following an arbitrary distribution. These results confirm recent experimental/theoretical findings that decreasing the nuclear export rate buffers the noise in mRNA level, and counterintuitively, decreasing the noise in the nuclear retention time enhances the noise in the mRNA level. Next, we further generalize the model to consider a dynamic extrinsic disturbance that affects the nuclear-to-cytoplasm export. Our results show that noise in the mRNA level varies non-monotonically with the disturbance timescale. More specifically, high- and low-frequency external disturbances have little impact on the mRNA noise level, while noise is amplified at intermediate frequencies. In summary, our results systematically uncover how the coupling of bursty transcription with nuclear export can both attenuate or amplify noise in mRNA levels depending on the nuclear retention time distribution and the presence of extrinsic fluctuations.

Performance Analysis of Work Stealing in Large-scale Multithreaded Computing

Randomized work stealing is used in distributed systems to increase performance and improve resource utilization. In this article, we consider randomized work stealing in a large system of homogeneous processors where parent jobs spawn child jobs that can feasibly be executed in parallel with the parent job. We analyse the performance of two work stealing strategies: one where only child jobs can be transferred across servers and the other where parent jobs are transferred. We define a mean-field model to derive the response time distribution in a large-scale system with Poisson arrivals and exponential parent and child job durations. We prove that the model has a unique fixed point that corresponds to the steady state of a structured Markov chain, allowing us to use matrix analytic methods to compute the unique fixed point. The accuracy of the mean-field model is validated using simulation. Using numerical examples, we illustrate the effect of different probe rates, load, and different child job size distributions on performance with respect to the two stealing strategies, individually, and compared to each other.

On the transmission of data packets through fiber-optic cables of uniform index

The treatment of Maxwell equations show that propagating wave of packets in fiber-optic cables is dispersive, propagating in groups, such that group velocity along certain curves in the frequency-phase velocity diagrams vanishes. It is suggested that stalling of wave groups is responsible, for bursting propagation observed in experimental measurements, causing some delay in transmission. The dispersion equations developed here, are different from those given in texts that are based on “weakly guiding approximation”. The queue of such data packets arriving at a router station is found to have a “raised tail” distribution unlike that of Poisson arrivals. For accounting the property, a Mittag–Leffler function distribution (MLFD) of probability is used following a modification of that for a Poisson process, the tail raising is shown to cause delay in transmission, and its estimate is analysed based on the theory.

Assigning Priorities (or Not) in Service Systems with Nonlinear Waiting Costs

For a queueing system with multiple customer types differing in service-time distributions and waiting costs, it is well known that the cµ-rule is optimal if costs for waiting are incurred linearly with time. In this paper, we seek to identify policies that minimize the long-run average cost under nonlinear waiting cost functions within the set of fixed priority policies that only use the type identities of customers independently of the system state. For a single-server queueing system with Poisson arrivals and two or more customer types, we first show that some form of the cµ-rule holds with the caveat that the indices are complex, depending on the arrival rate, higher moments of service time, and proportions of customer types. Under quadratic cost functions, we provide a set of conditions that determine whether to give priority to one type over the other or not to give priority but serve them according to first-come, first-served (FCFS). These conditions lead to useful insights into when strict (and fixed) priority policies should be preferred over FCFS and when they should be avoided. For example, we find that, when traffic is heavy, service times are highly variable, and the customer types are not heterogenous, so then prioritizing one type over the other (especially a proportionally dominant type) would be worse than not assigning any priority. By means of a numerical study, we generate further insights into more specific conditions under which fixed priority policies can be considered as an alternative to FCFS. This paper was accepted by Baris Ata, stochastic models and simulation.

Age of information of a server with energy requirements

We investigate a system with Poisson arrivals to two queues. One queue stores the status updates of the process of interest (or data packets) and the other handles the energy that is required to deliver the updates to the monitor. We consider that the energy is represented by packets of discrete unit. When an update ends service, it is sent to the energy queue and, if the energy queue has one packet, the update is delivered successfully and the energy packet disappears; however, in case the energy queue is empty, the update is lost. Both queues can handle, at most, one packet and the service time of updates is exponentially distributed. Using the Stochastic Hybrid System method, we characterize the average Age of Information of this system. Due to the difficulty of the derived expression, we also explore approximations of the average Age of Information of this system.

On Uniform Exponential Ergodicity of Markovian Multiclass Many-Server Queues in the Halfin–Whitt Regime

We study ergodic properties of Markovian multiclass many-server queues that are uniform over scheduling policies and the size of the system. The system is heavily loaded in the Halfin–Whitt regime, and the scheduling policies are work conserving and preemptive. We provide a unified approach via a Lyapunov function method that establishes Foster–Lyapunov equations for both the limiting diffusion and the prelimit diffusion-scaled queuing processes simultaneously. We first study the limiting controlled diffusion and show that if the spare capacity (safety staffing) parameter is positive, the diffusion is exponentially ergodic uniformly over all stationary Markov controls, and the invariant probability measures have uniform exponential tails. This result is sharp because when there is no abandonment and the spare capacity parameter is negative, the controlled diffusion is transient under any Markov control. In addition, we show that if all the abandonment rates are positive, the invariant probability measures have sub-Gaussian tails regardless whether the spare capacity parameter is positive or negative. Using these results, we proceed to establish the corresponding ergodic properties for the diffusion-scaled queuing processes. In addition to providing a simpler proof of previous results in Gamarnik and Stolyar [Gamarnik D, Stolyar AL (2012) Multiclass multiserver queueing system in the Halfin-Whitt heavy traffic regime: asymptotics of the stationary distribution. Queueing Systems 71(1–2):25–51], we extend these results to multiclass models with renewal arrival processes, albeit under the assumption that the mean residual life functions are bounded. For the Markovian model with Poisson arrivals, we obtain stronger results and show that the convergence to the stationary distribution is at an exponential rate uniformly over all work-conserving stationary Markov scheduling policies.